currentdensity

Current density, denoted J, is a vector field that represents the amount of electric current flowing per unit area at a point in a material or region. It points in the direction of the flow of positive charges and is related to the total current I by I = ∫ S J · dA over a cross-sectional surface S; for a uniform cross section, I = J A, so J = I/A. Its units are amperes per square meter (A/m^2).

In many materials, J is proportional to the electric field E. In homogeneous isotropic Ohmic materials, J = σE, where σ is the electrical conductivity. More generally, J = σ · E uses a conductivity tensor σ to account for anisotropy or nonlinear behavior. A microscopic view in metals gives J = nqμE, with n the carrier density, q the charge, and μ the mobility.

Charge conservation is described by the continuity equation ∂ρ/∂t + ∇·J = 0, where ρ is the charge density. In steady state, ∂ρ/∂t = 0 and ∇·J = 0, meaning current is locally conserved. Boundary conditions at interfaces relate the components of J to charges and fields; surface current density K (A/m) describes current confined to a surface and is tied to the tangential component of J.

J can vary in space and time and, in electromagnetism, contributes to Ampere’s law via ∇×B = μ0(J + ε0 ∂E/∂t). In materials science and device physics, J is often treated with tensor relations and drift–diffusion models that include both drift under E and diffusion of carriers.

Current density is a fundamental concept in electronics, electromagnetism, and material science, used to analyze conduction, power dissipation, and device performance.